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ECON10005-无代写
时间：2024-05-14
Subject Code: ECON10005
Subject Name: Quantitative Methods 1
QM1 Business Report
Submission: Draft
Tutorial Day and Time: Thursday, 9Am
Student ID Number Student Name
1. 1089251 Stacey Jade Li Tung Sang
2. 1172741 Georgia Marin
3. 1091892 Minsathi Chenuri Meegahapola
4.
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Report 1
This report will provide a holistic comparison of the property markets within the suburbs of
Richmond and South Yarra across 2018-2019. Firstly, descriptive statistics will be used to
analyse fluctuation in price as well as which suburb is on average more expensive overall.
Subsequently, frequency of property listings within respective price ranges in 2018 will be
evaluated based on property type and suburb. Then, the 2019 property market will be
analysed based on median price and correlation between price and proximity to primary
schools. Lastly descriptive statistics for 2018 and 2019 will allow for further evaluation.
Thus, clients with general interest in both areas will be able to decipher which suburb and
property are best suited.
TABLE 1.1- Descriptive Statistics on the overall property market
Table 1.1 demonstrates that average property price is higher in Richmond than in South
Yarra which is further supported by higher mode and median values. However, when
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investigating the prices separately per year and per type of housing it can be denoted that
South Yarra is actually more expensive than Richmond. This is conveyed through a higher
standard deviation, which suggests prices fluctuate more within South Yarra. The proportion
of bedrooms between properties are relatively the same for both suburbs. On average
properties in South Yarra are in closer proximity to primary schools than Richmond.
HISTOGRAMS BASED ON PROPERTY TYPE AND SUBURB
2018 Analysis
The above histograms convey there is a larger number of lower priced apartments in
Richmond in the ($300,000-$400,000) whereas majority of houses tend to have a higher
price ($1,000,000-$1,250,000) suggesting houses are more valued than apartments.
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In contrast, majority of apartment listings in South Yarra had a higher price range ($500,000-
$750,000), and houses were again more valued by buyers having majority of listings fall
between the ranges of ($1,250,000-$1,500,000 and $2,000,000-$2,250,000). Majority of
South Yarra houses were more expensive than any other property signifying prospective
buyers rate this area and property type the most.
TABLE 1.2
TABLE 1.3
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2019 Analysis
As seen in table 1.2 and 1.3 the 2019 market trends signify South Yarra was most expensive
in relation to 2-3-bedroom houses and 2-bedroom apartments, implying the area is
inhabited more by small families or groups. Whereas Richmond 1- and 3-bedroom
apartments were priced higher and was also the only suburb to list 4-bedroom apartments
suggesting individual’s living alone, couples ,big families or groups value this suburb more.
There was a strong positive association in the correlation between the distance between
primary school and price of houses in both suburbs apartment price may have increased
with further distance from a primary school, perhaps implying buyers value less disturbance
surrounding their property.
TABLE 1.4-Descriptive Statistics for Richmond (2018-2019)
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TABLE 1.5- Descriptive Statistics for South Yarra (2018-2019)
Table 1.4 and table 1.5 convey that the average house and apartment prices were higher in
South Yarra than in Richmond for both years which is supported by higher mode and
median values. In 2018, 25% (Q1) of the total houses were priced at $1,352,500 or lower for
South Yarra and $1,062,500 or lower for Richmond. Apartment prices in Richmond range
from $295,000 to $1,895,000 in 2019. The variation in house prices tend to be higher in
2019 compared to 2018 for both Richmond and South Yarra. In summary, when
investigating the prices separately per year and per type of housing it can be denoted that
South Yarra is actually more expensive than Richmond.
Moreover, it can be deciphered that mean house price and the proportion of apartments
differ between Richmond and South Yarra as seen in the data above and the conducted
hypothesis tests. Houses are typically more expensive than apartments in both Richmond
and South Yarra, however South Yarra is on average the more expensive suburb to buy
either property type in.
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Report 2
This report analyses the real estate industry, to identify and provide recommendations for a
client who is a first-time home buyer seeking for an apartment in two suburbs, either South
Yarra or Richmond with a budget of $500,000. Firstly, this report will compare the apartment
prices between the two suburbs. Then, we will consider which suburb has more options. Next,
we will examine the prices of apartments, number of bedrooms and distance from primary
school for each suburb, to finally describe the correlation of prices and distance to primary
school and market trends and test whether
apartments are more expensive in
Richmond than South Yarra.
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PRICES (000S)
Histogram for apartment prices in
Richmond
Figure 1.1
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200220240260280300320340360380400420440460480500
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PRICES (000S)
Historgram for apartment prices in
South Yarra
Figure 1.2
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From the histogram above, Fig 1.1 has a unimodal illustrating that most of the apartment prices
in Richmond fall under $400,000. Fig 1.2 also has a unimodal suggesting that most of the prices
in South Yarra are $360,000. This highlights that there are cheaper options available in South
Yarra. Considering the proportions in Fig 2.1, we state that there are also more available
options in South Yarra than in Richmond.
Prices in
Richmond
($000s) Bedrooms
Distance of
Primary school
from
Richmond/km
Mean 394.04
0.430
Mode 395.0 1 0.310
Median 395.0 1 0.350
Max 487.5 2 0.840
Min 310.0 1 0.240
SD 51.18
0.164
Prices in South
Yarra($000s) Bedrooms
Distance of
Primary school
from South
Yarra/km
Mean 400.7
0.361
Mode 345.0 1 0.110
Median 402.5 1 0.340
Max 500.0 2 0.790
Min 280.0 1 0.110
SD 70.11
0.184
Figure 2.1
Table 2.2 Table 2.3
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On both Table 2.2 and 2.3, prices in South Yarra on average seem more expensive than in Richmond.
However, since they’re more available options in Richmond than in South Yarra the value of the mean
isn’t fair. There are 2 bedrooms’ choices available in both Suburbs. The standard deviation establishes
that prices in South Yarra are more volatile than in Richmond. If distance from primary school is of
interest to the client, South Yarra proves to be the closer with an average of 0.361km and a mode of
0.11km compared to an average of 0.430km and a mode of 0.310km in Richmond.
The covariance establishes that there is a positive relationship between prices and distance from
primary school, meaning that as prices go up, schools are further away. However, the correlation
coefficient from fig 3.3, 3.1 and 3.2 state that the relationship is weak especially for Richmond.
Highlighting that people are willing to pay more to not be close to schools, perhaps they prefer
tranquility.
Figure 3.1 Figure 3.2
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The time-series plot figure 4.0 gives us information about the market trend for apartments in the two
suburbs in the year 2018. Apartment prices tend to fluctuate rapidly over time. At a 5% significance level
there isn’t enough evidence that states whether Richmond has higher apartment prices than South
Yarra.
To conclude, we can view that there are more available options to choose from in South Yarra. Despite
Fig 1.2 depicts that most of the apartments cost around $360,000, on average prices are more expensive
in South Yarra than in Richmond. On average schools are closer to apartments in South Yarra than in
Richmond and most of the apartments constitute of only 1 bedroom.
Figure 4.0
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Report 3
This report is specifically written for a client who are a dual income family with a young child.
Firstly, this report will compare the type of of properties available in Richmond that is, both
houses and apartments under $1,000,000. Then we will be making price comparisons between
the apartments and houses available in this suburb. Next, Since the client has a young child, we
will also be focusing our attention on data relevant to the correlation/covariance between
prices of apartments/houses and the distance to primary schools as the proximity to primary
schools, might be an important factor they will be taking into consideration when selecting a
home. Finally, we’ve also given some brief information about the market trend for apartments
in 2018 and 2019.
This report is written specifically for a client whose looking for a House/Apartment in Richmond
who has a budget of $100,000,000. 15% of the properties available under this budget are
Figure 3.1
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houses while the rest of the 85% of the properties available are apartments. Therefore, there
are more apartments available under this budget and this is supported by the pie chart (figure
3.1) given above.
It can clearly be observed from the table 3.1 that, two-bedroom houses and apartments seem
to be the most common type available under this budget. The average house price is $884,500
and for apartments its $532,010. Median Apartment prices seem to be lower than that of the
houses as well. Therefore, apartments seem to be generally lower in price compare to houses
and this is indicated by both the mean and median prices stated above. In addition to this
apartment might be more convenient for this type of family.
Given that they have a young child, the distance to primary schools seems to be an important
factor the buyers would be considering when looking for a home. As seen in the table 3.1
above, on average the mean distance to primary schools from houses and apartments is,
0.4814km and 0.4983km respectively. Therefore, both houses and apartments seem to be
located in close proximity to primary schools in Richmond.
Table 3.1
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As in in the table above, (table 3.2) the covariance between house/apartment prices and
distance to primary schools indicates a positive relationship between the two variables. There
seems to be a positive correlation coefficient as well, thereby indicating that as the distance to
primary schools increases and prices of houses and apartments also increases and vice versa.
However, the correlation values seen in table 3.2 suggests that, the relationship that exists
between house/apartment prices and the distance to primary schools is very weak even though
it is positive. This is also supported by figures 3.2 and 3.3 given below, which suggests that the
relationship between the two variables is not strong.
Therefore, In Richmond, House and Apartment prices are likely to be higher if they are situated
further away from primary schools. As the client, has a young child they are more likely to buy
an apartment or house closer to primary schools and in this case, it will cost them much lesser if
the property was closer to the schools.
Correlation Covariance
House Prices vs Distance to Primary Schools 0.0715 1.1073
Apartment Prices vs Distance to Primary Schools 0.2909 9.7046
Table 3.2
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Figure 3.4, shows the fluctuations in apartment prices in 2018 and 2019. Prices tend to be
extremely volatile and therefore there’s no clear pattern. However, given the data, the prices
Figure 3.4
Figure 3.2 Figure 3.3
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tend to be rising towards the end of 2019 and towards the end of 2018, the prices seem to be
experiencing a sharp fall.
Overall, it can be deducted that the client has more options available in apartments than
houses in Richmond as suggested by the data above and the hypothesis test (conducted in
the appendix) shows that the average prices of houses more expensive than the average
prices of apartments.
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REPORT 1-Hypothesis test 1
TEST OF DIFFERENCE BETWEEN MEAN HOUSE PRICES IN RICHMOND AND SOUTH YARRA
Are the mean house prices of the same in Richmond and South Yarra, or are they different?
Let, Mu1=Mean house price in Richmond
Mu2=Mean house price in South Yarra
H0: Mu1=Mu2 = Mu1=0
HA: Mu1≠Mu2 = Mu1≠0
Significance level(alpha)=5%
Xbar1-Xbar2= 593.2225
S.E(Xbar1-Xbar2) =sqrt (se (Xbar1) ^2+se(Xbar2) ^2) = -17.0114
t=(Xbar1-Xbar2)/s.e(Xbar1-Xbar2) = 2.4698
Richmond SouthYarra Difference
n 85 135
mean 593.2225 635.2372 -42.0147
s.d 269.3613 537.1176
s.e 29.2163 46.2277 -17.0114
t stat 2.4698
dof 84 134 218
pval.2t 0.0143
95% C.I upper 651.32 726.67
95% C.I lower 535.12 543.81
Critical Value(CV) 1.9886 1.9778 0.0108
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Degrees of freedom= (85+135)-2=218
pval.2t=T.DIST.2T (ABS (2.4698),218) = 0.0143
Since p<0.05 Reject H0.
Conclusion: Mean house price in Richmond is not equal to mean house price in South Yarra.
Therefore, the means are different and there is evidence that there are statistically different
to each other.
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Calculating Confidence Intervals
The confidence intervals show the extent to which you have confidence in the mean value
(precision of the mean)
• Richmond
Xbar1= $593.2225
Degrees of Freedom=85-1=84
Critical Value (CV)= T.INV (0.975,84) = 1.9886
Confidence Interval= Xbar1+/- tcv*(s.e(Xbar1) = (535.12,651.32)
If price falls within this confidence interval DO NOT reject H0.
Average price of a house in Richmond is $593.2225 with a 95% Confidence interval of
(535.123,651.32)
• South Yarra
Xbar2=$635.2372
Degrees of Freedom=135-1=134
Critical Value (CV)= T.INV (0.975,134) = 1.9778
Confidence Interval= Xbar1+/-C. V*(s.e(Xbar1) = (543.81, 726.67)
Any value that (price of a house) falls within this confidence interval DO NOT reject H0. Any
value (price of a house) outside this Confidence level, REJECT H0.
Average price of a house in South Yarra is $635.2372 with a 95% Confidence interval of
(543.807, 726.668).
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Report 1- Hypothesis Test 2
TEST OF DIFFERENCE BETWEEN PROPORTION OF APARTMENTS IN RICHMOND AND
SOUTH YARRA
Are the proportions of apartments the same in South Yarra and Richmond, or
are they different?
Let, Mu1 = proportion of apartments in Richmond
Mu2= proportions of apartments in South Yarra
H0: Mu1=Mu2 = Mu1=0
HA: Mu1≠Mu2 = Mu1≠0
Significance level(alpha)=5%
1̅̅̅̅ = 85168 = 0.506 2̅̅̅̅ = 135158 = 0.854 . (1) = √0.506×(1−0.506)168 . (2) = √0.854×(1−0.854)158
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Conclusion: Since p<0.05 we reject the null hypothesis and conclude that the proportions are
different in both suburbs.
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Report 2-Hypotheis Test
ARE APARTMENTS MORE EXPENSIVE IN RICHMOND THAN IN SOUTH YARRA ?
Mu1=average apartment price in Richmond
Mu2=average apartment price in South Yarra
H0:Mu1=Mu2=0
HA: Mu1>Mu2
Significance level(alpha)=5%
RICHMOND SOUTH
YARRA
Difference/Combined
Number of
Apartments
85 135
Average price X 593.22 635.237 42.017
Standard
deviation
269.3612 537.117
Standard error 29.216 46.228 29.2162+46.2282=54.686
Degrees of
freedom
84 134 218
t value
42.017/54.686=0.768
p value
T. DIST.RT(0.768,218) = 0.222
Since p > 0.05, do not reject H0
In this case the evidence does not suggest
Richmond apartments are more expensive than
South yarra
Critical value
test
t=0.768
t/2, n1+n2-2=1.9709
Decision rule: reject H0 if t > t/2, n1+n2-2
0.768 < 1.9709, therefore don’t reject
There is evidence, at the 5% significance level
that apartments are not more expensive in
Richmond than South Yarra
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95% Confidence interval = [-63.764, 151.798]
Therefore, the estimated difference between Richmond and South Yarra prices is 215.5618
Conclusion: The evidence does not suggest that Richmond has higher apartment prices
than South Yarra.
95%
Confidence
interval lower
T.INV (1-0.025,218) =1.9709056
44.017-1.9709056*54.686
= -63.764
95%
Confidence
interval upper
44.017+1.9709056*54.686
=151.798
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Report 3-Hypothesis Test
Are prices of houses on average more expensive than apartments?
Let, Mu1= mean of House prices in Richmond
Mu2= mean of prices for apartments in Richmond
H0: Mu1=Mu2 = Mu1=0
HA: Mu1>Mu2 = Mu1>0
Significance level(alpha)=5%
Conclusion: Since p<0.05 we reject the null hypothesis and conclude that prices of houses are
on average more expensive than the average prices of apartments.
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Subject Name: Quantitative Methods 1
QM1 Business Report
Submission: Draft
Tutorial Day and Time: Thursday, 9Am
Student ID Number Student Name
1. 1089251 Stacey Jade Li Tung Sang
2. 1172741 Georgia Marin
3. 1091892 Minsathi Chenuri Meegahapola
4.
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Report 1
This report will provide a holistic comparison of the property markets within the suburbs of
Richmond and South Yarra across 2018-2019. Firstly, descriptive statistics will be used to
analyse fluctuation in price as well as which suburb is on average more expensive overall.
Subsequently, frequency of property listings within respective price ranges in 2018 will be
evaluated based on property type and suburb. Then, the 2019 property market will be
analysed based on median price and correlation between price and proximity to primary
schools. Lastly descriptive statistics for 2018 and 2019 will allow for further evaluation.
Thus, clients with general interest in both areas will be able to decipher which suburb and
property are best suited.
TABLE 1.1- Descriptive Statistics on the overall property market
Table 1.1 demonstrates that average property price is higher in Richmond than in South
Yarra which is further supported by higher mode and median values. However, when
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investigating the prices separately per year and per type of housing it can be denoted that
South Yarra is actually more expensive than Richmond. This is conveyed through a higher
standard deviation, which suggests prices fluctuate more within South Yarra. The proportion
of bedrooms between properties are relatively the same for both suburbs. On average
properties in South Yarra are in closer proximity to primary schools than Richmond.
HISTOGRAMS BASED ON PROPERTY TYPE AND SUBURB
2018 Analysis
The above histograms convey there is a larger number of lower priced apartments in
Richmond in the ($300,000-$400,000) whereas majority of houses tend to have a higher
price ($1,000,000-$1,250,000) suggesting houses are more valued than apartments.
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In contrast, majority of apartment listings in South Yarra had a higher price range ($500,000-
$750,000), and houses were again more valued by buyers having majority of listings fall
between the ranges of ($1,250,000-$1,500,000 and $2,000,000-$2,250,000). Majority of
South Yarra houses were more expensive than any other property signifying prospective
buyers rate this area and property type the most.
TABLE 1.2
TABLE 1.3
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2019 Analysis
As seen in table 1.2 and 1.3 the 2019 market trends signify South Yarra was most expensive
in relation to 2-3-bedroom houses and 2-bedroom apartments, implying the area is
inhabited more by small families or groups. Whereas Richmond 1- and 3-bedroom
apartments were priced higher and was also the only suburb to list 4-bedroom apartments
suggesting individual’s living alone, couples ,big families or groups value this suburb more.
There was a strong positive association in the correlation between the distance between
primary school and price of houses in both suburbs apartment price may have increased
with further distance from a primary school, perhaps implying buyers value less disturbance
surrounding their property.
TABLE 1.4-Descriptive Statistics for Richmond (2018-2019)
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TABLE 1.5- Descriptive Statistics for South Yarra (2018-2019)
Table 1.4 and table 1.5 convey that the average house and apartment prices were higher in
South Yarra than in Richmond for both years which is supported by higher mode and
median values. In 2018, 25% (Q1) of the total houses were priced at $1,352,500 or lower for
South Yarra and $1,062,500 or lower for Richmond. Apartment prices in Richmond range
from $295,000 to $1,895,000 in 2019. The variation in house prices tend to be higher in
2019 compared to 2018 for both Richmond and South Yarra. In summary, when
investigating the prices separately per year and per type of housing it can be denoted that
South Yarra is actually more expensive than Richmond.
Moreover, it can be deciphered that mean house price and the proportion of apartments
differ between Richmond and South Yarra as seen in the data above and the conducted
hypothesis tests. Houses are typically more expensive than apartments in both Richmond
and South Yarra, however South Yarra is on average the more expensive suburb to buy
either property type in.
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Report 2
This report analyses the real estate industry, to identify and provide recommendations for a
client who is a first-time home buyer seeking for an apartment in two suburbs, either South
Yarra or Richmond with a budget of $500,000. Firstly, this report will compare the apartment
prices between the two suburbs. Then, we will consider which suburb has more options. Next,
we will examine the prices of apartments, number of bedrooms and distance from primary
school for each suburb, to finally describe the correlation of prices and distance to primary
school and market trends and test whether
apartments are more expensive in
Richmond than South Yarra.
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PRICES (000S)
Histogram for apartment prices in
Richmond
Figure 1.1
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200220240260280300320340360380400420440460480500
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Historgram for apartment prices in
South Yarra
Figure 1.2
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From the histogram above, Fig 1.1 has a unimodal illustrating that most of the apartment prices
in Richmond fall under $400,000. Fig 1.2 also has a unimodal suggesting that most of the prices
in South Yarra are $360,000. This highlights that there are cheaper options available in South
Yarra. Considering the proportions in Fig 2.1, we state that there are also more available
options in South Yarra than in Richmond.
Prices in
Richmond
($000s) Bedrooms
Distance of
Primary school
from
Richmond/km
Mean 394.04
0.430
Mode 395.0 1 0.310
Median 395.0 1 0.350
Max 487.5 2 0.840
Min 310.0 1 0.240
SD 51.18
0.164
Prices in South
Yarra($000s) Bedrooms
Distance of
Primary school
from South
Yarra/km
Mean 400.7
0.361
Mode 345.0 1 0.110
Median 402.5 1 0.340
Max 500.0 2 0.790
Min 280.0 1 0.110
SD 70.11
0.184
Figure 2.1
Table 2.2 Table 2.3
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On both Table 2.2 and 2.3, prices in South Yarra on average seem more expensive than in Richmond.
However, since they’re more available options in Richmond than in South Yarra the value of the mean
isn’t fair. There are 2 bedrooms’ choices available in both Suburbs. The standard deviation establishes
that prices in South Yarra are more volatile than in Richmond. If distance from primary school is of
interest to the client, South Yarra proves to be the closer with an average of 0.361km and a mode of
0.11km compared to an average of 0.430km and a mode of 0.310km in Richmond.
The covariance establishes that there is a positive relationship between prices and distance from
primary school, meaning that as prices go up, schools are further away. However, the correlation
coefficient from fig 3.3, 3.1 and 3.2 state that the relationship is weak especially for Richmond.
Highlighting that people are willing to pay more to not be close to schools, perhaps they prefer
tranquility.
Figure 3.1 Figure 3.2
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The time-series plot figure 4.0 gives us information about the market trend for apartments in the two
suburbs in the year 2018. Apartment prices tend to fluctuate rapidly over time. At a 5% significance level
there isn’t enough evidence that states whether Richmond has higher apartment prices than South
Yarra.
To conclude, we can view that there are more available options to choose from in South Yarra. Despite
Fig 1.2 depicts that most of the apartments cost around $360,000, on average prices are more expensive
in South Yarra than in Richmond. On average schools are closer to apartments in South Yarra than in
Richmond and most of the apartments constitute of only 1 bedroom.
Figure 4.0
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Report 3
This report is specifically written for a client who are a dual income family with a young child.
Firstly, this report will compare the type of of properties available in Richmond that is, both
houses and apartments under $1,000,000. Then we will be making price comparisons between
the apartments and houses available in this suburb. Next, Since the client has a young child, we
will also be focusing our attention on data relevant to the correlation/covariance between
prices of apartments/houses and the distance to primary schools as the proximity to primary
schools, might be an important factor they will be taking into consideration when selecting a
home. Finally, we’ve also given some brief information about the market trend for apartments
in 2018 and 2019.
This report is written specifically for a client whose looking for a House/Apartment in Richmond
who has a budget of $100,000,000. 15% of the properties available under this budget are
Figure 3.1
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houses while the rest of the 85% of the properties available are apartments. Therefore, there
are more apartments available under this budget and this is supported by the pie chart (figure
3.1) given above.
It can clearly be observed from the table 3.1 that, two-bedroom houses and apartments seem
to be the most common type available under this budget. The average house price is $884,500
and for apartments its $532,010. Median Apartment prices seem to be lower than that of the
houses as well. Therefore, apartments seem to be generally lower in price compare to houses
and this is indicated by both the mean and median prices stated above. In addition to this
apartment might be more convenient for this type of family.
Given that they have a young child, the distance to primary schools seems to be an important
factor the buyers would be considering when looking for a home. As seen in the table 3.1
above, on average the mean distance to primary schools from houses and apartments is,
0.4814km and 0.4983km respectively. Therefore, both houses and apartments seem to be
located in close proximity to primary schools in Richmond.
Table 3.1
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As in in the table above, (table 3.2) the covariance between house/apartment prices and
distance to primary schools indicates a positive relationship between the two variables. There
seems to be a positive correlation coefficient as well, thereby indicating that as the distance to
primary schools increases and prices of houses and apartments also increases and vice versa.
However, the correlation values seen in table 3.2 suggests that, the relationship that exists
between house/apartment prices and the distance to primary schools is very weak even though
it is positive. This is also supported by figures 3.2 and 3.3 given below, which suggests that the
relationship between the two variables is not strong.
Therefore, In Richmond, House and Apartment prices are likely to be higher if they are situated
further away from primary schools. As the client, has a young child they are more likely to buy
an apartment or house closer to primary schools and in this case, it will cost them much lesser if
the property was closer to the schools.
Correlation Covariance
House Prices vs Distance to Primary Schools 0.0715 1.1073
Apartment Prices vs Distance to Primary Schools 0.2909 9.7046
Table 3.2
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Figure 3.4, shows the fluctuations in apartment prices in 2018 and 2019. Prices tend to be
extremely volatile and therefore there’s no clear pattern. However, given the data, the prices
Figure 3.4
Figure 3.2 Figure 3.3
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tend to be rising towards the end of 2019 and towards the end of 2018, the prices seem to be
experiencing a sharp fall.
Overall, it can be deducted that the client has more options available in apartments than
houses in Richmond as suggested by the data above and the hypothesis test (conducted in
the appendix) shows that the average prices of houses more expensive than the average
prices of apartments.
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REPORT 1-Hypothesis test 1
TEST OF DIFFERENCE BETWEEN MEAN HOUSE PRICES IN RICHMOND AND SOUTH YARRA
Are the mean house prices of the same in Richmond and South Yarra, or are they different?
Let, Mu1=Mean house price in Richmond
Mu2=Mean house price in South Yarra
H0: Mu1=Mu2 = Mu1=0
HA: Mu1≠Mu2 = Mu1≠0
Significance level(alpha)=5%
Xbar1-Xbar2= 593.2225
S.E(Xbar1-Xbar2) =sqrt (se (Xbar1) ^2+se(Xbar2) ^2) = -17.0114
t=(Xbar1-Xbar2)/s.e(Xbar1-Xbar2) = 2.4698
Richmond SouthYarra Difference
n 85 135
mean 593.2225 635.2372 -42.0147
s.d 269.3613 537.1176
s.e 29.2163 46.2277 -17.0114
t stat 2.4698
dof 84 134 218
pval.2t 0.0143
95% C.I upper 651.32 726.67
95% C.I lower 535.12 543.81
Critical Value(CV) 1.9886 1.9778 0.0108
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Degrees of freedom= (85+135)-2=218
pval.2t=T.DIST.2T (ABS (2.4698),218) = 0.0143
Since p<0.05 Reject H0.
Conclusion: Mean house price in Richmond is not equal to mean house price in South Yarra.
Therefore, the means are different and there is evidence that there are statistically different
to each other.
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Calculating Confidence Intervals
The confidence intervals show the extent to which you have confidence in the mean value
(precision of the mean)
• Richmond
Xbar1= $593.2225
Degrees of Freedom=85-1=84
Critical Value (CV)= T.INV (0.975,84) = 1.9886
Confidence Interval= Xbar1+/- tcv*(s.e(Xbar1) = (535.12,651.32)
If price falls within this confidence interval DO NOT reject H0.
Average price of a house in Richmond is $593.2225 with a 95% Confidence interval of
(535.123,651.32)
• South Yarra
Xbar2=$635.2372
Degrees of Freedom=135-1=134
Critical Value (CV)= T.INV (0.975,134) = 1.9778
Confidence Interval= Xbar1+/-C. V*(s.e(Xbar1) = (543.81, 726.67)
Any value that (price of a house) falls within this confidence interval DO NOT reject H0. Any
value (price of a house) outside this Confidence level, REJECT H0.
Average price of a house in South Yarra is $635.2372 with a 95% Confidence interval of
(543.807, 726.668).
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Report 1- Hypothesis Test 2
TEST OF DIFFERENCE BETWEEN PROPORTION OF APARTMENTS IN RICHMOND AND
SOUTH YARRA
Are the proportions of apartments the same in South Yarra and Richmond, or
are they different?
Let, Mu1 = proportion of apartments in Richmond
Mu2= proportions of apartments in South Yarra
H0: Mu1=Mu2 = Mu1=0
HA: Mu1≠Mu2 = Mu1≠0
Significance level(alpha)=5%
1̅̅̅̅ = 85168 = 0.506 2̅̅̅̅ = 135158 = 0.854 . (1) = √0.506×(1−0.506)168 . (2) = √0.854×(1−0.854)158
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Conclusion: Since p<0.05 we reject the null hypothesis and conclude that the proportions are
different in both suburbs.
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Report 2-Hypotheis Test
ARE APARTMENTS MORE EXPENSIVE IN RICHMOND THAN IN SOUTH YARRA ?
Mu1=average apartment price in Richmond
Mu2=average apartment price in South Yarra
H0:Mu1=Mu2=0
HA: Mu1>Mu2
Significance level(alpha)=5%
RICHMOND SOUTH
YARRA
Difference/Combined
Number of
Apartments
85 135
Average price X 593.22 635.237 42.017
Standard
deviation
269.3612 537.117
Standard error 29.216 46.228 29.2162+46.2282=54.686
Degrees of
freedom
84 134 218
t value
42.017/54.686=0.768
p value
T. DIST.RT(0.768,218) = 0.222
Since p > 0.05, do not reject H0
In this case the evidence does not suggest
Richmond apartments are more expensive than
South yarra
Critical value
test
t=0.768
t/2, n1+n2-2=1.9709
Decision rule: reject H0 if t > t/2, n1+n2-2
0.768 < 1.9709, therefore don’t reject
There is evidence, at the 5% significance level
that apartments are not more expensive in
Richmond than South Yarra
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95% Confidence interval = [-63.764, 151.798]
Therefore, the estimated difference between Richmond and South Yarra prices is 215.5618
Conclusion: The evidence does not suggest that Richmond has higher apartment prices
than South Yarra.
95%
Confidence
interval lower
T.INV (1-0.025,218) =1.9709056
44.017-1.9709056*54.686
= -63.764
95%
Confidence
interval upper
44.017+1.9709056*54.686
=151.798
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Report 3-Hypothesis Test
Are prices of houses on average more expensive than apartments?
Let, Mu1= mean of House prices in Richmond
Mu2= mean of prices for apartments in Richmond
H0: Mu1=Mu2 = Mu1=0
HA: Mu1>Mu2 = Mu1>0
Significance level(alpha)=5%
Conclusion: Since p<0.05 we reject the null hypothesis and conclude that prices of houses are
on average more expensive than the average prices of apartments.
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